DOI: http://dx.doi.org/10.18273/revint.v34n2-2016006

Un algoritmo cuasi-Newton para resolver
la ecuación cuadrática matricial

MAURICIO MACÍAS^a*, HÉCTOR J. MARTÍNEZ^b,
ROSANA PÉREZ^a

^a Universidad del Cauca, Departamento de Matemáticas, Popayán, Colombia.
_b Universidad del Valle, Departamento de Matemáticas, Cali, Colombia.

Resumen. En este artículo proponemos un algoritmo cuasi-Newton para resolver una ecuación cuadrática matricial, el cual reduce el costo computacional del método Newton-Schur, tradicionalmente usado para resolver dicha ecuación. Demostramos que el algoritmo propuesto es local y hasta cuadráticamente convergente. Presentamos pruebas numéricas que ratifican los resultados teóricos desarrollados.

Palabras clave: Función cuadrática matricial, operador de Fréchet diferenciable, método de Newton-Schur, convergencia cuadrática. ]]> MSC2010: 65H10, 49M15, 90C53, 15A24, 39B42.

A quasi-Newton algorithm to solve the matrix
quadratic equation

Abstract. In this paper we propose a quasi-Newton algorithm to solve a matrix quadratic equation, which reduces the computational cost of Newton-Schur method, traditionally used to solve this equation. We show that the proposed algorithm is local and up to quadratically convergent. We present some numerical tests which confirm the theoretical results developed.

Keywords: matrix cuadratic equation, Fréchet derivative operator, Newton-Schur method, quasi-Newton method, cuadratic convergence.

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Referencias

[1] Bai Z-Z., Guo X-X and Yin J-F., "On two iteration methods for the quadratic matrix equations", Inter. J. Numer. Anal. Model. 2 (2005), suppl., 144-122.         [ Links ]

[2] Berhanu M., "The polynomial eigenvalue problem", Thesis (Ph.D.), University of Manchester, 2005, 219 p.         [ Links ]

[3] Bermúdez A., Durán R.G., Rodríguez R. and Solomin J., "Finite element analysis of a quadratic eingenvalue problem arising in dissipative acoustics", SIAM J. Numer. Anal. 38 (2000), No. 1, 267-291.         [ Links ]

[4] Butler G.J., Johnson C.R. and Wolkowicz H., "Nonnegative solutions of a quadratic matrix equation arising from comparison theorems in ordinary differential equations", SIAM J. Algebraic Discrete Methods, 6 (1985), No. 1, 47-53.         [ Links ]

[5] Clark J.V., Zhou N. and Pister K.S.J., "Modified nodal analysis for MEMS with multienergy domains", International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, San Diego, CA, March, 2000.         [ Links ]

[6] Dávila C.E., "A subspace approach to estimation of autoregressive parameters from noisy measurements", IEEE Trans. Signal Process. 46 (1998), No. 2, 531-534.         [ Links ]

[7] Davis G.J., "Numerical solution of a quadratic matrix equation", SIAM J. Sci. Statist. Comput. 2 (1981), No. 2, 164-175.         [ Links ]

[8] Dennis J.E., Jr., and Schnabel R.B., Numerical methods for unconstrained optimization and nonlinear equations. Prentice Hall, Inc., Englewood Cliffs, NJ, 1983.         [ Links ]

[9] Dennis J.E., Jr., Traub J.F. and Weber R.P., "The algebraic theory of matriz polynomials", SIAM J. Numer. Anal. 13 (1976), No. 6, 831-845.         [ Links ]

[10] Gao Y-H., "Newton's method for the quadratic matrix equation", Appl. Math. Comput. 182 (2006), No. 2, 1772-1779.         [ Links ]

[11] Golub G.H. and Van Loan C.F., Matrix computations, Third ed. Jonhs Hopkins University Press, Baltimore, MD, 1996.         [ Links ]

[12] Golub G.H., Nash S. and Van Loan C.F., "A Hessenberg-Schur method for the problem AX+XB=C", IEEE Trans. Automat. Control. 24 (1979), No. 6, 909-913.         [ Links ]

[13] Guo C-H., "On a quadratic matrix equation associated whit an M-matrix", IMA J. Numer. Anal. 23 (2003), No. 1, 11-27.         [ Links ]

[14] Higham N.J., Functions of matrices. Theory and computations, SIAM, Philadelphia, PA, 2008.         [ Links ]

[15] Higham N.J., "Stable iterations for the matrix square root", Numer. Algorithms 15 (1997), No. 2, 227-242.         [ Links ]

[16] Higham N.J. and Kim H-M., "Numerical analysis of a quadratic matrix equation", IMA J. Numer. Anal. 20 (2000), No. 4, 499-519.         [ Links ]

[17] Higham N.J. and Kim H-M., "Solving a quadratic matrix equation by Newton's method with exact line searches", SIAM J. Matrix Anal. Appl. 23 (2001), No. 2, 303-316.         [ Links ]

[18] Horn R.A. and Johnson C.R., Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.         [ Links ]

[19] Kay S.M., "Noise compensation for autoregressive spectral estimates", IEEE Trans. Acoust. Speech Signal Process. ASSP-28 (1980), No. 3, 292-303.         [ Links ]

[20] Lancaster P. Lambda-matrices and vibrating systems, Pergamon Press, Oxford-New York-París, 1966.         [ Links ]

[21] Laub A.J., "Effient multivariable frequency response computations", 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes, 39-41, December, 1980.         [ Links ]

[22] Liu L-D., "Perturbation analysis of a quadratic matrix equation associated with an M-matrix", J. Comput. Appl. Math. 260 (2014), 410-419.         [ Links ]

[23] Liu L-D. and Lu X., "Two kinds of condition numbers for the quadratic matrix equation", Appl. Math. Comput. 219 (2013), No. 16, 8759-8769.         [ Links ]

[24] Macías E.M., "Métodos secantes de cambio mínimo para el cálculo de ceros de funciones de matrices", Tesis de maestría, Universidad del Cauca, 2013.         [ Links ]

[25] Macías M., Martínez H.J. and Pérez R., "Sobre la convergencia de un método secante para ecuaciones matriciales no lineales", Rev. Integr. Temas Mat. 32 (2014), No. 2, 181-197.         [ Links ]

[26] Monsalve M. and Raydan M., "A secant method for nonlinear matrix problems", in Numerical linear algebra in signals, systems and control, Springer (2011), 387-402.         [ Links ]

[27] Monsalve M. and Raydan M., "Newton's method and secant methods: a longstanding relationship from vectors to matrices", Port. Math. 68 (2011), No. 4, 431-475.         [ Links ]

[28] Reddy S.C., Schmid P.J. and Henningson D.S., "Pseudospectra of the Orr-Sommerfeld operator", SIAM J. App. Math. 53 (1993), No. 1, 15-47.         [ Links ]

[29] Seo S-H., Seo J-H. and Kim H-M , "Newton's method for solving a quadratic matrix equation whith special coeficient matrices", Honam Math. J. 35 (2013), No. 3, 417-433.         [ Links ]

[30] Smith H.A., Singh R.K. and Sorensen D.C., "Formulation and solution of the non-linear, damped eigenvalue problem for skeletal systems", Internat. J. Numer. Methods Engrg. 38 (1995), No. 18, 3071-3085.         [ Links ]

[31] Tisseur F., "Backward error and condition of polynomial eigenvalue problems", Linear Algebra Appl. 309 (2000), No. 1-3, 339-361.         [ Links ]

[32] Tisseur F. and Meerbergen K., "The quadratic eigenvalue problem", SIAM Rev. 43 (2001), No. 2, 235-286.         [ Links ]

[33] Thomson W.T., Theory of vibration with applications, Fourth ed., CRC Press, 1996.         [ Links ]

[34] Watkins D.S., Fundamentals of matrix computations. Second ed, Pure and Applied Mathematics, Wiley-Interscience, New York, 2002.         [ Links ]

[35] Xu H.G. and Lu L.Z., "Properties of a quadratic matrix equation and the solution of the continuos-time algebraic Riccati equation", Linear Algebra Appl. 222 (1995), 127-145.         [ Links ]

[36] Zhou N., Clark J.V. and Pister K.S.J., "Nodal simulation for MEMS design using SUGAR v0.5", International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, Santa Clara, CA, 308-313, 1988.         [ Links ]

^*E-mail: mauroma@unicauca.edu.co ]]> Recibido: 01 de julio de 2016, Aceptado: 17 de noviembre 2016.
Para citar este artículo: M. Macías, H.J. Martínez, R. Pérez, Un algoritmo cuasi-Newton para resolver la
ecuación cuadrática matricial, Rev. Integr. Temas Mat. 34 (2016), No. 2, 187-206.